Assuming we know of both organizations above that their contributions were necessary, both can claim having helped 600,000 chickens, without needing to help 1,200,000 chickens in total.
This problem cannot be mitigated by thinking probabilistically. If there is probability p_s_A (p_s_B) of organisation A (B) being successful acting alone, p_s of organisations A and B being successful acting together, p_A (p_B) of organisation A (B) acting, and impact N given success, the expected counterfactual value of:
A acting is CV_A = ((1 - p_B)*p_s_A + p_B*p_s—p_B*p_s_B)*N.
B acting is CV_B = ((1 - p_A)*p_s_B + p_A*p_s—p_A*p_s_A)*N.
The sum of the expected counterfactual values of A and B is CV = CV_A + CV_B = ((1 - p_A—p_B)*(p_s_A + p_s_B) + (p_A + p_B)*p_s)*N. This can be as large as 2*N when A and B can never succeed alone (p_s_A, p_s_B = 0), A and B always succeed acting together (p_s = 1), and A and B are certain to act (p_A, p_B = 1).
The problem is solved using Shapley values. The expected Shapley value of:
A acting is SV_A = ((1 - p_B)*p_s_A + p_B*p_s/​2 - p_B*p_s_B)*N.
B acting is SV_B = ((1 - p_A)*p_s_B + p_A*p_s/​2 - p_A*p_s_A)*N.
The sum of the expected Shapley values of A and B is SV = SV_A + SV_B = ((1 - p_A—p_B)*(p_s_A + p_s_B) + (p_A + p_B)/​2*p_s)*N. This can only be as large as N when A and B can never succeed alone (p_s_A, p_s_B = 0), A and B always succeed acting together (p_s = 1), and A and B are certain to act (p_A, p_B = 1).
This problem cannot be mitigated by thinking probabilistically. If there is probability p_s_A (p_s_B) of organisation A (B) being successful acting alone, p_s of organisations A and B being successful acting together, p_A (p_B) of organisation A (B) acting, and impact N given success, the expected counterfactual value of:
A acting is CV_A = ((1 - p_B)*p_s_A + p_B*p_s—p_B*p_s_B)*N.
B acting is CV_B = ((1 - p_A)*p_s_B + p_A*p_s—p_A*p_s_A)*N.
The sum of the expected counterfactual values of A and B is CV = CV_A + CV_B = ((1 - p_A—p_B)*(p_s_A + p_s_B) + (p_A + p_B)*p_s)*N. This can be as large as 2*N when A and B can never succeed alone (p_s_A, p_s_B = 0), A and B always succeed acting together (p_s = 1), and A and B are certain to act (p_A, p_B = 1).
The problem is solved using Shapley values. The expected Shapley value of:
A acting is SV_A = ((1 - p_B)*p_s_A + p_B*p_s/​2 - p_B*p_s_B)*N.
B acting is SV_B = ((1 - p_A)*p_s_B + p_A*p_s/​2 - p_A*p_s_A)*N.
The sum of the expected Shapley values of A and B is SV = SV_A + SV_B = ((1 - p_A—p_B)*(p_s_A + p_s_B) + (p_A + p_B)/​2*p_s)*N. This can only be as large as N when A and B can never succeed alone (p_s_A, p_s_B = 0), A and B always succeed acting together (p_s = 1), and A and B are certain to act (p_A, p_B = 1).